For this Assignment the Submission Instructions will be slightly different. In addition to the single pdf with your handwritten calculations and results, you will also have to submit:

1. All the source files and or scripts needed to generate your results. So you can submit Python (-py), MATLAB (m) or Gnuplot (plt) scripts, Mathematica (.nb) or Jupyter (. ipynb) notebooks. Be especially careful with Jupyter notebooks. I should be able to execute your submitted files and reproduce your results.
2. Calculus handwritten required

Consider the bivariate objective function f(x,y)=x^(2)+y^(2)+xy-5x-7y+20. You are to calculate the exact relative (and absolute) minimum by hand as well as approximate it using gradient descent. For more information, read the relevant lecture notes and textbook sections. Specifically, you are to : (a) Plot f(x,y) on -2 <= x <= 4,-2 <= y <= 4. (Code 8 image file with plot) (10 pts) (b) Use Calculus to find the exact point x_(min)=(x_(min),y_(min)) at which f attains its relative minimum, as well as its minimum value f_(min). Why is this also the absolute minimum? (Scanned handwritten) (20 pts) (c) Formulate gradient descent for the objective function f. The learning rate should be alpha=0.4 and the initial point x^([0])=(x^([0]),y^([0]))=(2,2). (Scanned handwritten)

(d) Write and execute your own code in order to implement the algorithm you formulated in

(c). The convergence criterion is | f(x^[k+1])) - f(x^[k])| < 10^(-5) where x^(k) = (x^[k], y^[k]) is the

k-th gradient descent iterate and fk = f(x^(k)) is the value of the objective function at that

point. (Code & screenshot of output)

Your code must correctly implement the algorithm (20 pts) be well-commented (10 pts)

and should output:

i. The approximation x* = (x*, y*) of the location of the minimum.

ii. The corresponding value f* = f(x*) of the objective function.

iii. The iterate number n at which convergence occurs.

(20 pts)

(5 pts)

(5 pts)

(e) Use a computational utility and a ready routine command to confirm your results. (Code & screenshot)

(5 pts)

(f Plot the values of the objective function fk as a function of the iteration number k. Note

that k = 0, 1,2,. ..,n, where n is the iteration number at which convergence is reached.

(Code & image file with plot)

(10 pts)